Solve 0.85=1{0.5}^frac{t{15.05}}

$$0.85 = 1 { 0.5 }^{ \frac{ t }{ 15.05 } }$$

Solve for t

$$t=\frac{301\log_{2}\left(\frac{5}{17}\right)}{20}+30.1\approx 3.528702067$$

Steps Using Definition of Logarithm

Divide both sides by 1.

$$\frac{0.85}{1}=0.5^{\frac{t}{15.05}}$$

Expand \frac{0.85}{1} by multiplying both numerator and the denominator by 100.

$$\frac{85}{100}=0.5^{\frac{t}{15.05}}$$

Reduce the fraction \frac{85}{100} to lowest terms by extracting and canceling out 5.

$$\frac{17}{20}=0.5^{\frac{t}{15.05}}$$

Swap sides so that all variable terms are on the left hand side.

$$0.5^{\frac{t}{15.05}}=\frac{17}{20}$$

Use the rules of exponents and logarithms to solve the equation.

$$0.5^{\frac{20}{301}t}=0.85$$

Take the logarithm of both sides of the equation.

$$\log(0.5^{\frac{20}{301}t})=\log(0.85)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\frac{20}{301}t\log(0.5)=\log(0.85)$$

Divide both sides by \log(0.5).

$$\frac{20}{301}t=\frac{\log(0.85)}{\log(0.5)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$\frac{20}{301}t=\log_{0.5}\left(0.85\right)$$

Divide both sides of the equation by \frac{20}{301}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=-\frac{\frac{\ln(\frac{17}{20})}{\ln(2)}}{\frac{20}{301}}$$